A shock absorbing bumper adaptable for mounting onto the front and rear of a vehicle such as an automobile, the bumper comprising an elastic external member having sheathed therein a flexible core material and a decorative facing provided on the front side of the external member. An elastic facing base having formed on its front side the elastic decorative facing is provided on its rear side with a hook or hooks which fit into the corresponding slits formed in the front portion of the external member and are engaged therein in such a way that they overhang the edges of the slits while the rims of the facing base fit in the lateral notches also formed in the external member.